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Use symmetry to graph the polar curve and identify the type of curve. r=3+3cos theta

Use symmetry to graph the polar curve and identify the type of curve.\newliner=3+3cos⁡θ r=3+3 \cos \theta

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Convert complex numbers between rectangular and polar formright chevron icon

Full solution

Q. Use symmetry to graph the polar curve and identify the type of curve.\newliner=3+3cos⁡θ r=3+3 \cos \theta
  1. Plot key points: Plot key points. Start with θ=0\theta=0. r(0)=3+3cos⁡(0)=3+3(1)=6r(0) = 3 + 3\cos(0) = 3 + 3(1) = 6.
  2. Check symmetry: Check symmetry. Since cos⁡(θ)=cos⁡(−θ)\cos(\theta) = \cos(-\theta), the curve is symmetric about the polar axis (x-axis).
  3. Find more points: Find more points. Let's do θ=π2\theta=\frac{\pi}{2}. r(π2)=3+3cos⁡(π2)=3+3(0)=3r(\frac{\pi}{2}) = 3 + 3\cos(\frac{\pi}{2}) = 3 + 3(0) = 3.
  4. Continue with: Continue with θ=π\theta=\pi. r(π)=3+3cos⁡(π)=3+3(−1)=0r(\pi) = 3 + 3\cos(\pi) = 3 + 3(-1) = 0.
  5. Check: Check θ=3π2\theta=\frac{3\pi}{2}. r(3π2)=3+3cos⁡(3π2)=3+3(0)=3r\left(\frac{3\pi}{2}\right) = 3 + 3\cos\left(\frac{3\pi}{2}\right) = 3 + 3(0) = 3.
  6. Plot the points: Plot the points and draw the curve smoothly connecting them, considering the symmetry.
  7. Identify the type: Identify the type of curve. It's a limaçon with an inner loop because r(π)=0r(\pi) = 0 and r(0)>r(π2)r(0) > r(\frac{\pi}{2}).

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